Awesome, thanks for the great tips!
It was 20* outside, I went out and both cars fired right up like nobody's business.
I pulled the cap on the '81 - oddly, written on the outside of the cap in black magic marker was, "Must be grey in color."
Okay....
The cap looked fine and the inside was bone dry with no bad-looking stuff anywhere. I cleaned the contacts with a small wire brush, sprayed it down with WD-40, wiped it off and cleaned the rotor as well (it looked just fine).
I put the cap back on and it started with no issues which was no surprise because it's always started after it's warmed up.
I checked voltage at the coil and when running it was 8.94-ish.
I turned the car off, turned the ignition on and connected my remote start switch to the solenoid and the positive cable of the battery, hit the button and......no start. It'd crank just fine but it wouldn't start.
I hit the key and it fired right up.
I did this a few times and the engine won't start unless I use the key - except for one time.
Just when I let off the remote start button, it caught and fired up but somehow, somewhere, there's got to be a circuit that's completed when you crank the engine with the key that's NOT completed when you just put power to the solenoid.
So, at least I know that trying to start it with the remote start button connected the way it's connected isn't going to work for me.
Next question - where do I connect the remote start button's leads to allow me to start the car with the ignition switch in the "Run" position?
I think this will lead me to the problem. I think that whatever circuit is completed via the ignition switch being turned to the "Start/Crank" position isn't being completed when it's damp. I dunno why, I dunno how but I'm thinking that's where the issue lies.....
Also, I don't know if I mentioned it but the coils and ignition modules are new as well. I cleaned all leads and connectors before connecting them.
Thanks again for everybody's ideas here!
Dave